240 ANALYTICAL MECHANICS external and those which are internal to the system. Let F/ denote the resultant of the external forces acting on the particle, and F/' denote the resultant of the internal forces acting upon it, due to its connection with the rest of the system. Then F<=F/+F<". But since F is the resultant of all the external forces acting upon all the particles of the system we have F=2F/ The second sum of the left-hand member is the sum of the internal forces and is nil, because the internal forces come in pairs which mutually annul each other. Therefore F = 2Fi (IV) = Smv (IV} = |(2mv). (V) These are results which are worth noting. Equation (IV) states that the resultant external force acting upon a system equals and is opposite to the vector sum (or the resultant) of the kinetic reactions of all the particles of the system. Equation (V) states that the resultant external force acting upon a system equals the time rate of change of the resultant momentum of the system. PROBLEMS. (1) Show that the component, along any direction, of the resultant force acting upon a particle equals the rate at which the corresponding component of its momentum changes, that is, X = -r(mx), etc. Cut (2) Show that the component, along any direction, of the resultant external force acting upon a system equals the rate at which the corre-